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92416a2 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Computability.UniqueGames.Decoding.ActualSeedSamplingLemmas
import OAI.Computability.UniqueGames.Machines.MachineSingleOrbitBridgeLemmas
import OAI.Computability.UniqueGames.PCP.BinaryParityReduction
import OAI.Computability.UniqueGames.Reduction.FinishReduction
namespace OAI
namespace UniqueGamesTheorem.FromMatrixGap
open UniqueGamesTheorem.Foundations Target
open UniqueGamesTheorem.Foundations.Complexity
open UniqueGamesTheorem.Integration
open UniqueGamesTheorem.Reduction
open Explicit
noncomputable section
/-- Choose the parity error only after the matrix tuple length is fixed, then
compose the actual raw-input, matrix-test, and final graph machines. -/
def reduction {ε δ : ℝ} (P : ParameterSelection.OuterParameters ε δ)
(M : Decoder.MatrixGap.Parameters P.pStar) :
MachineOutputContract.BinaryGapReduction ε δ := by
classical
have hk : 0 < M.k := M.k_pos
let ξ := Classical.choose (P.exists_parity_error M.k hk)
have hξspec := Classical.choose_spec (P.exists_parity_error M.k hk)
have hξ : 0 < ξ := hξspec.1
have hξsmall : ξ < 1 / 100 := hξspec.2.1
have hξbudget : ξ < P.epsilon0 / (2 * M.k * P.repetitions) := hξspec.2.2
let R := Classical.choice (Outer.BinaryParityReduction.exists_reduction ξ hξ hξsmall)
let matrix := fun input => MachineSingleOrbitBridge.output M.k M.T (R.reduce input)
let matrixMachine := MachineSequential.composeBits
(f := R.reduce) (g := MachineSingleOrbitBridge.output M.k M.T)
R.computation (MachineSingleOrbitBridge.computableInPolyTime M.k M.T)
have matrixFinite : MachineFiniteAlphabet.FiniteAlphabet matrixMachine.tm :=
MachineFiniteAlphabet.composeBits
(f := R.reduce) (g := MachineSingleOrbitBridge.output M.k M.T)
R.computation (MachineSingleOrbitBridge.computableInPolyTime M.k M.T)
R.finiteAlphabet (MachineSingleOrbitBridge.workAlphabetFinite M.k M.T)
have hq : 2 ≤ 2 ^ M.s := Nat.le_self_pow (by have h := M.s_pos; omega) 2
refine FinishReduction.of_matrix_reduction P hq M.s_pos
(Encoding.alphabetEquiv M.s).symm matrix ?_ matrixMachine matrixFinite
((M.k : ℚ) * ξ + P.pStar / 2)
(P.matrix_error_budget M.k hk ξ hξbudget.le) ?_ ?_
· intro input
exact M.output_translations (Outer.HastadSource.asSource (R.reduce input))
· intro input yes
obtain ⟨assignment, hfailure⟩ :=
Outer.BinaryParityReduction.failure_completeness R input yes
have h := M.completeness (Outer.HastadSource.asSource (R.reduce input))
assignment ξ hfailure
simpa only [matrix, MachineSingleOrbitBridge.output,
MachineSingleOrbitBridge.SingleAddress.tableOutput, Outer.HastadSource.asSource,
Rat.cast_add, Rat.cast_mul, Rat.cast_natCast,
Rat.cast_div, Rat.cast_ofNat] using h
· intro input no
exact M.sound (Outer.HastadSource.asSource (R.reduce input))
(R.distinct input) (Outer.BinaryParityReduction.parity_soundness R input no)
end
end UniqueGamesTheorem.FromMatrixGap
end OAI
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